Question

A police bike with its siren on, moving with a velocity $v_s$ toward a man standing at a bus stop. As the bike approaches, the man hears the sound at a frequency of $f_d$. Take the speed of sound to be $v$.
Calculate the wavelength of the sound produced by the siren?

• A. $\frac{v-v_s}{v{f}_d}$
• B. $\frac{v^2}{(v+v_s)f_d}$
• C. $\frac{v^2}{(v-v_s)f_d}$
• D. $\frac{v+v_s}{v{f}_d}$
• E. $\frac{v^2{f}_d}{v-v_s}$

Answer 1

Answer: (C)

$\frac{v^2}{(v-v_s)f_d}$

Let the original frequency of the siren (source) be $f$.

Apparent frequency heard by the man (observer) $=f_d$

Using Doppler effect in case when the source moves towards the stationary observer :

$\therefore$ $f_d=f[\frac{v}{v-v_s}]$ $⟹f=\frac{v-v_s}{v}{f}_d$

Thus wave length of the sound produced by siren $\lambda =\frac{v}{f}=\frac{v^2}{(v-v_s)f_d}$